[tex]\sqrt{5x-1}-\sqrt{x}=1\\\\\left(\sqrt{5x-1}\right)^2=\left(\sqrt{x}+1\right)^2\\\\5x-1=x+2\sqrt{x}+1\\\\4x-2=2\sqrt{x}\;\;\div(2\\\\\left(2x-1\right)^2=\left(\sqrt{x}\right)^2\\\\4x^2-4x+1=x\\\\4x^2-5x+1=0[/tex]
Calculando as raízes encontramos [tex]\boxed{S=\left\{\frac{1}{4},1\right\}}[/tex]
Por fim, devemos VERIFICAR se as raízes satisfazem a equação irracional inicial, veja:
- Quando x = 1:
[tex]\sqrt{5x-1}-\sqrt{x}=1\\\\\sqrt{5-1}-\sqrt{1}=1\\\\\sqrt{4}-\sqrt{1}=1\\\\2-1=1\\\\1=1\Rightarrow\text{verdadeira}[/tex]
- Quando x = 1/4
[tex]\sqrt{5x-1}-\sqrt{x}=1\\\\\sqrt{\frac{5}{4}-1}-\sqrt{\frac{1}{4}}=1\\\\\sqrt{\frac{1}{4}}-\sqrt{\frac{1}{4}}=1\\\\0=1\Rightarrow\text{falsa}[/tex]
Logo, a resposta final é [tex]\boxed{\boxed{x=1}}[/tex]
